Question

A facilities manager at a university reads in a research report that the mean amount of time spent in the shower by an adult is 5 minutes. He decides to collect data to see if the mean amount of time that college students spend in the shower is significantly different from 5 minutes. In a sample of 8 students, he found the average time was 5.55 minutes and the standard deviation was 0.75 minutes. Using this sample information, conduct the appropriate hypothesis test at the 0.05 level of significance. Assume normality.

Answer 1

Answer:

0.0384 < 0.05, which means that we can conclude that the mean amount of time that college students spend in the shower is significantly different from 5 minutes.

Step-by-step explanation:

A facilities manager at a university reads in a research report that the mean amount of time spent in the shower by an adult is 5 minutes.

This means that the null hypothesis is $H_{0} = 5$

He decides to collect data to see if the mean amount of time that college students spend in the shower is significantly different from 5 minutes.

This means that the alternate hypothesis is $H_{a} \neq 5$

The test statistic is:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

In which X is the sample mean, $\mu$ is the value tested at the null hypothesis, $\sigma$ is the standard deviation and n is the size of the sample.

Null hypothesis:

Tests $H_{0} = 5$, which means that $\mu = 5$

In a sample of 8 students, he found the average time was 5.55 minutes and the standard deviation was 0.75 minutes.

This means, respectively, that $n = 8, \mu = 5.55, \sigma = 0.75$

Test statistic:

The test statistic is:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

$z = \frac{5.55 - 5}{\frac{0.75}{\sqrt{8}}}$

$z = 2.07$

Pvalue:

Since we are testing if the mean is different from a value, and z is positive. The pvalue is two multiplied by 1 subtracted by the pvalue of z = 2.07.

z = 2.07 has a pvalue of 0.9808

2*(1 - 0.9808) = 2*(0.0192) = 0.0384

Decision:

0.0384 < 0.05, which means that we can conclude that the mean amount of time that college students spend in the shower is significantly different from 5 minutes.